Parsimonious truncated Newton method for time-domain full waveform inversion based on Fourier-domain full-scattered-field approximation

In order to exploit Hessian information in Full Waveform Inversion (FWI), the matrix-free truncated Newton method can be used. In such a method, Hessian-vector product computation is one of the major concerns due to the huge memory requirements and demanding computational cost. Using the adjoint-state method, the Hessian-vector product can be estimated by zero-lag cross-correlation of the first-order/second-order incident wavefields and the second-order/first-order adjoint wavefields. Different from the implementation in frequency-domain FWI, Hessian-vector product construction in the time domain becomes much more challenging as it is not affordable to store the entire time-dependent wavefields. The widely used wavefield recomputation strategy leads to computationally intensive tasks. We present an efficient alternative approach to computing the Hessian-vector product for time-domain FWI. In our method, discrete Fourier transform is applied to extract frequency-domain components of involved wavefields, which are used to compute wavefield cross-correlation in the frequency domain. This makes it possible to avoid reconstructing the first-order and second-order incident wavefields. In addition, a full-scattered-field approximation is proposed to efficiently simplify the second-order incident and adjoint wavefields computation, which enables us to refrain from repeatedly solving the first-order incident and adjoint equations for the second-order incident and adjoint wavefields (re)computation. With the proposed method, the computational time can be reduced by 70% and 80% in viscous media for Gauss-Newton and full-Newton Hessian-vector product construction, respectively. The effectiveness of our method is also verified in the frame of a 2D multi-parameter inversion, in which the proposed method almost reaches the same iterative convergence of the conventional time-domain implementation.

A New Hyper-Parameter Optimization Method for Power Load Forecast Based on Recurrent Neural Networks

The selection of the hyper-parameters plays a critical role in the task of prediction based on the recurrent neural networks (RNN). Traditionally, the hyper-parameters of the machine learning models are selected by simulations as well as human experiences. In recent years, multiple algorithms based on Bayesian optimization (BO) are developed to determine the optimal values of the hyper-parameters. In most of these methods, gradients are required to be calculated. In this work, the particle swarm optimization (PSO) is used under the BO framework to develop a new method for hyper-parameter optimization. The proposed algorithm (BO-PSO) is free of gradient calculation and the particles can be optimized in parallel naturally. So the computational complexity can be effectively reduced which means better hyper-parameters can be obtained under the same amount of calculation. Experiments are done on real world power load data,where the proposed method outperforms the existing state-of-the-art algorithms,BO with limit-BFGS-bound (BO-L-BFGS-B) and BO with truncated-newton (BO-TNC),in terms of the prediction accuracy. The errors of the prediction result in different models show that BO-PSO is an effective hyper-parameter optimization method.

Elastic full-waveform inversion of steeply dipping structures with prismatic waves

High-resolution reconstruction of steeply dipping structures is an important but challenging subject in seismic exploration. Prismatic reflections that contain information on these structures are helpful for reconstructing steeply dipping structures. Elastic full-waveform inversion (EFWI) is a powerful tool that can accurately estimate subsurface parameters from multicomponent seismic data, which can provide information useful for characterizing oil and gas reservoirs. We construct the relationship between the forward and inverse problems related to the prismatic reflections by considering the multiparameter exact Hessian in realistic elastic media. We numerically analyze the characteristics of the multiparameter exact Hessian and show that when prismatic reflections are apparent in multicomponent data, the multiparameter delta Hessian has a strong influence. We explain this in more detail through the forward analysis and demonstrate that the multiparameter delta Hessian considers not only the prismatic reflections but also compensates for the primary reflections in multicomponent data. To use the prismatic waves, we develop a migration/demigration approach-based truncated Newton (TN) method in frequency-domain EFWI, whose storage requirements and the computational costs are the same as those of the truncated Gauss–Newton (TGN) method. Realistic 2D numerical examples demonstrate that, compared with TGN method based on the first-order Born approximation, the TN method can converge faster and obtain higher accuracy in the reconstruction of steeply dipping structures.

Assessment of seismic ambient noise parameter estimation and wavefield decomposition in WaveDec

<p>Knowledge about seismic ambient noise wavefield through decomposition into different participant waves is of special importance in geophysical studies. In this study, WaveDec technique (Maranò et al., 2012) as an array statistical signal processing technique was used to decompose seismic ambient noise wavefield and to estimate wavefield parameters. In this method, the measurements from all components of stations and parameters of interest are modeled jointly which leads to significant improvement in extracting characteristics of surface waves. Considering the contribution of both Love and Rayleigh waves in the wavefield, the method estimates the desired parameters including amplitude, phase, azimuth, wave number and the ellipticity angle (for the Rayleigh wave only) based on the Maximum Likelihood Estimation method. One of the main characteristic of WaveDec is estimating the ellipticity angle of Rayleigh waves. This is very beneficial in determining retrograde and prograde particle motion and also in mode distinction.</p><p>In the WaveDec algorithm, the Truncated Newton method is used to optimize likelihood functions with respect to wavefield parameters. Furthermore, Bayesian Information Criterion (BIC) is used to select the best model and wave type determination (Rayleigh, Love, body wave or noise). Regarding a group of generated models for different wave types, the one with the smallest BIC is chosen.</p><p>We examined consistency of WaveDec algorithm by applying different numerical optimization methods; Truncated Newton, L-BFGS-B quasi-Newton and simplex-based Nelder-Mead methods. Furthermore, different model selection criteria; BIC, Akaike Information Criterion (AIC) and Hannan–Quinn Information Criterion (HQC) were examined to study the quality of generated models. They possess different penalty terms to avoid overfitting the models on data. All possible pairs of optimization methods and model selection criteria were utilized and replaced in WaveDec algorithm. In order to compare the resultant dispersion curves of surface waves and ellipticity angle curves of Rayleigh waves, SESAME model M2.1 synthetic data and some seismic ambient noise measurements in Colfiorito basin in Italy (Array B) were analyzed.</p>

Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation

In this work, we deal with Truncated Newton methods for solving large scale (possibly nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix decomposition, which is well exploited in the widely adopted SYMMBK (Bunch and Kaufman in Math Comput 31:163–179, 1977; Chandra in Conjugate gradient methods for partial differential equations, vol 129, 1978; Conn et al. in Trust-region methods. MPS-SIAM series on optimization, Society for Industrial Mathematics, Philadelphia, 2000; HSL, A collection of Fortran codes for large scale scientific computation, http://www.hsl.rl.ac.uk/; Marcia in Appl Numer Math 58:449–458, 2008) routine. It can be used to provide conjugate directions, both in the case of $$1\times 1$$ 1 × 1 and $$2\times 2$$ 2 × 2 pivoting steps. The main drawback is that the resulting solution of Newton’s equation might not be gradient–related, in the case the objective function is nonconvex. Here we first focus on some theoretical properties, in order to ensure that at each iteration of the Truncated Newton method, the search direction obtained by using an adapted Bunch and Kaufman factorization is gradient–related. This allows to perform a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the results of an extended numerical experience using large scale CUTEst problems is reported, showing the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems.

Control and Optimization of Interfacial Flows Using Adjoint-Based Techniques

The applicability of adjoint-based gradient computation is investigated in the context of interfacial flows. Emphasis is set on the approximation of the transport of a characteristic function in a potential flow by means of an algebraic volume-of-fluid method. A class of optimisation problems with tracking-type functionals is proposed. Continuous (differentiate-then-discretize) and discrete (discretize-then-differentiate) adjoint-based gradient computations are formulated and compared in a one-dimensional configuration, the latter being ultimately used to perform optimisation in two dimensions. The gradient is used in truncated Newton and steepest descent optimisers, and the algorithms are shown to recover optimal solutions. These validations raise a number of open questions, which are finally discussed with directions for future work.

Use of prismatic waves in full-waveform inversion with the exact Hessian

The truncated Newton method uses information contained in the exact Hessian in full-waveform inversion (FWI). The exact Hessian physically contains information regarding doubly scattered waves, especially prismatic events. These waves are mainly caused by the scattering at steeply dipping structures, such as salt flanks and vertical or nearly vertical faults. We have systematically investigated the properties and applications of the exact Hessian. We begin by giving the formulas for computing each term in the exact Hessian and numerically analyzing their characteristics. We show that the second term in the exact Hessian may be comparable in magnitude to the first term. In particular, when there are apparent doubly scattered waves in the observed data, the influence of the second term may be dominant in the exact Hessian and the second term cannot be neglected. Next, we adopt a migration/demigration approach to compute the Gauss-Newton-descent direction and the Newton-descent direction using the approximate Hessian and the exact Hessian, respectively. In addition, we determine from the forward and the inverse perspectives that the second term in the exact Hessian not only contributes to the use of doubly scattered waves, but it also compensates for the use of single-scattering waves in FWI. Finally, we use three numerical examples to prove that by considering the second term in the exact Hessian, the role of prismatic waves in the observed data can be effectively revealed and steeply dipping structures can be reconstructed with higher accuracy.